Stephen C. Wales

An ensemble source spectra model for merchant ship-radiated noise

This paper presents an evaluation of the classical model for determining an ensemble of the broadband source spectra of the sound generated by individual ships and proposes an alternate model to overcome the deficiencies in the classical model. The classical model, proposed by Ross [Mechanics of Underwater Noise (Pergamon, New York, 1976)] postulates that the source spectrum for an individual ship is proportional to a baseline spectrum with the constant of proportionality determined by a power-law relationship on the ship speed and length. The model evaluation, conducted on an ensemble of 54 source spectra over a 30-1200-Hz to 1200-Hz frequency band, shows that this assumption yields large rms errors in the broadband source level for the individual ships and significantly overestimates the variability in the source level across the ensemble of source spectra. These deficiencies are a consequence of the negligible correlation between the source level and the ship speed and the source level and the ship length. The alternate model proposed here represents the individual ship spectra by a modified rational spectrum where the poles and zeros are restricted to the real axis and the exponents of the terms are not restricted to integer values. An evaluation of this model on the source spectra ensemble indicates that the rms errors are significantly less than those obtained with any model where the frequency dependence is represented by a single baseline spectrum. Furthermore, at high frequencies (400 to 1200 Hz), a single-term rational spectrum model is sufficient to describe the frequency dependence and, at the low frequencies (30 to 400 Hz), there is only a modest reduction in the rms error for a higher order model. Finally, a joint probability density on the two parameters of the single term model based on the measured histograms of these parameters is proposed. This probability density provides a mechanism for generating an ensemble of ship spectra.

Factorization and Path Integration of the Helmholtz Equation: Numerical Algorithms

The propagator for the reduced scalar Helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation. Path (functional) integrals are taken to provide the principal representation of the propagator and are computed directly. The path integral is the primary tool in extending the classical Fourier methods, so appropriate for wave propagation in homogeneous media, to inhomogeneous media. For transversely inhomogeneous environments, the n‐dimensional Helmholtz equation can be exactly factored into separate forward and backward one‐way wave equations. A parabolic‐based (one‐way) phase space path integral construction provides the generalization of the Tappert/Hardin split‐step FFT algorithm to the full one‐way (factored Helmholtz) wave equation. These extended marching algorithms can readily accommodate density profiles and range updating, and further, in conjunction with imbedding methods, provide the basis for incorporating backscatter effects. In a complementary manner, for general range‐dependent environments, elliptic‐based (two‐way) path integral constructions lead to an approximate representation of the propagator (Feynman/Garrod) and a natural statistical (Monte Carlo) means of evaluation. Taken together, the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations, resolution of the square root operator, and the concept of an underlying stochastic process. The one‐way marching algorithms are applied to ocean acoustic environments, seismological environments, and extreme model environments designed to establish their range of validity and manner of breakdown.

SHIPPING NOISE PREDICTIONS: CAPABILITIES AND LIMITATIONS

This paper presents an example of the impact on the noise of relying only on the Historical Temporal Shipping (HITS) model for a site near San Diego. The paper presents an argument against using the HITS model to predict trends in the speed and length of the worlds shipping fleet. Also presented is the example that illustrates the extent of the disparity in noise predictions that can occur due to differences in the acoustic propagation and the environmental model components of a noise model. An additional example illustrates the impact on the noise of underestimating the coastal shipping components. The paper presents and summarizes the results obtained.

Ambient noise vertical directionality in the northwest Atlantic

The vertical directionality of the ambient noise field was measured at a site between Cape Hatteras and Bermuda over the frequency range 45–100 Hz. The noise level was observed to decrease rapidly at angles beyond ±20 ° from the horizontal by as much as 20 dB. Slight peaks were observed near ±15 ° and at the horizontal. These peaks are hypothesized to be due to deep ocean shipping and slope enhanced signals from shipping located on the continental shelf edge, respectively. Theoretical calculations based on a parabolic equation propagation model and measured environmental information support these hypotheses.

Elements of a geoacoustic model of the upper crust

Elements of a geoacoustic model of the geologically young (< 10 million years) upper crust, the top few hundred meters of the basalt subbottom, are described. The model is based on analysis of low‐frequency reflectivity versus angle data at a 1 million‐year‐old sediment‐free site, as well as limited Deep Sea Drilling Project downhole logging results, and extrapolated seismic refraction measurements. The upper crust at this site, which is in close proximity to the crest of the East Pacific Rise, may be characterized by low interfacial velocities (Vp∼2800 m/s and Vs∼800 m/s), large gradients (3–5 s−1), and substantial rms roughness (∼5 m). The low‐inferred shear speed implies no shear critical angle. Hence, low‐frequency energy incident on the bottom at small grazing angles is in large part transmitted into the rock, refracted by the gradient, and reradiated into the water. Scattering loss occurs at both the initial incidence and the interaction of the refracted energy at the boundary. Low interfacial shear...

Phase space methods and path integration: the analysis and computation of scalar wave equations

Abstract The scalar Helmholtz equation plays a significant role in studies of electromagnetic, seismic, and acoustic direct wave propagation. Phase space, or ‘microscopic’, methods and path (functional) integral representations provide the appropriate framework to extend homogeneous Fourier methods to inhomogeneous environments. The two complementary approaches to this analysis and computation of the n -dimensional Helmholtz propagator are reviewed. For the factorization/(one-way) path integration/invariant imbedding approach, the exact solution of the Helmholtz composition equation for the Weyl square root operator symbol is presented in the quadratic case. The filtered, one-way, phase space marching algorithm is examined in detail and compared numerically with wide-angle, one-way, partial differential wave equations formally derived from approximation theory. For the second approach, which directly constructs approximate two-way path functionals, the feasibility of a Monte Carlo (statistical) evaluation of the Feynman/Garrod propagator is discussed.

A Vector Parabolic Equation Model for Elastic Propagation

Some aspects of vector parabolic equations are examined as a means of propagating waves in linear elastic solids. A revised derivation of a previously published model, based on a Born approximation for scattering from an inhomogeneous layer, enables parabolic equation (PE) computations in elastic media with different longitudinal and shear speeds. This method of derivation is shown to reduce to the normal PE for the scalar case. The elastic PE model has been implemented using finite-difference techniques. Computed results for a channeled environment are consistent with physical considerations. Analysis reveals that hard and soft boundaries do not lend themselves to a straightforward implementations in a potential formulation.

Concurrent inversion of geo- and bio-acoustic parameters from transmission loss measurements in the Yellow Sea.

This paper describes results of a simultaneous inversion of bio-acoustic parameters of fish (anchovies) and geo-acoustic parameters of the bottom from transmission loss (TL) measurements in the Yellow Sea, which were reported by Qiu et al. [J. Sound Vib. 220, 331-342 (1999)]. This data set was selected because the bio-absorptivity at their site was extremely large, 40 dB at 1.3 kHz at 5 km, and measurements were made between multiple source and receiver depths and ranges. Measurements were made at night when anchovies are generally dispersed. Replica fields were calculated with a normal mode model, which incorporates bio-absorption layers. The inversion was based on minimizing the rms difference, delta, between measured and calculated values of TL at all ranges and source and receiver depths, and involved a simultaneous search for bio-layer depth, bio-layer thickness, bio-alpha, geo-sound speed, and geo-alpha. The resultant small value of delta, +/- 1.7 dB, confirmed that the model, which was assumed in replica field calculations, was realistic, and that inverted parameters were meaningful. In particular, the inverted depth of the bio-absorption layer, 6.9 +/- 0.3 m, was consistent with theoretical calculations of the depth, 5.8 +/- 1 m, of 10-cm-long anchovies Engraulis japonicus, the dominant species in the Yellow Sea.

Very-low-frequency under-ice reflectivity

This paper describes a direct method to model under‐ice reflection loss from analysis of ice draft data taken from a region of the Arctic near the FRAM IV experiment site. The water–ice boundary is modeled as a random distribution of infinite elliptical half‐cylinders fixed to a free surface. Burke and Twersky’s theory [J. Acoust. Soc. Am. 40, 883–895 (1966)] of scattering from a single cylindrical protuberance is used to calculate reflectivity from a distribution of scatterers. Individual ice ridge keels are identified from ice draft data resulting in a ridge keel depth distribution function spatially coincident with FRAM IV acoustic propagation paths. The scattering amplitude is weighted by the keel depth distribution function providing an effective ridge keel depth which is used to calculate the under‐ice reflectivity. The predicted reflection loss is in good agreement with those inferred from normal‐mode methods applied to FRAM IV acoustic field data received on a larger aperture vertical array.

Effects of Upper Crustal Geoacoustic Parameters on Low Frequency Sound

A seismo-acoustic paradigm of the upper crust is proposed, based on analysis of low-frequency reflectivity versus angle data at a 0.5 million-year old sediment-free site, and on spatially well sampled seismic refraction measurements at sediment-free and thinly sedimented sites in the Pacific Ocean. Crustal velocity measurements at thick sediment covered sites are evidently substantially higher, and are excluded from the analysis presented here. Interfacial compressional and shear speeds of thinly sedimented, geologically young upper crust (between 0.5 and 5 million years) at this site are estimated to be about 2800 m/s and 800 m/s respectively. At such sites sub-basement gradients are approximately 4 s-1. The rms roughness of the basement, which is Fresnel zone size dependent, has significant effects at frequencies as low as 10 Hz. Low shear speeds imply no shear critical angle. Hence, at low shear speed sites, low-frequency energy incident on the bottom at small grazing angles is in large part transmitted into the rock, refracted by the gradient, and reradiated into the water. Scattering loss occurs at both the initial incidence and the interaction of the refracted energy at the boundary. Low interfacial shear speeds lead to large grazing angles at the boundary for the transmitted shear waves, large wavenumbers and, hence large boundary scattering losses; small changes in interfacial shear speed produce large changes in subsurface boundary-scattering loss, and hence in the reflection coefficient. Sediment-free/thinly sediment-covered crustal shear speeds are projected to increase with age, eventually (at an unknown age) becoming faster than the speed of sound in water, resulting in critical angle reflection.